Investigation of some basi̇c concepts of fractional order analysis

This master's thesis includes the main material, consisting of an introduction and four chapters, conclusions and a list of used literature sources. The introductory part of the work provides information from the literature related to the subject of the study, the relevance of the study, the purpose of the study and the tasks that need to be solved to achieve this goal. The second chapter provides a theoretical basis for the definition of derivatives and integrals of fractional order and for the study of their properties, as well as for the study of differential equations containing derivatives of fractional order. The third chapter defines the gamma and beta functions, which are commonly used in the calculus of fractional derivatives and integral calculus, and are known as the Euler functions, and presents their relationship properties. In the fourth chapter, the Riemann-Liouville, Grunwald-Letnikov, Caputo approximations used in fractional-order mathematics are given, and some examples of calculating the derivatives and the fractional-order integral using these approaches are given. In the fifth chapter, firstly, the connection between differential equations of fractional order and the Volterra integral equation of the second kind is determined, and then the existence and uniqueness of the solution of the Cauchy-type problem for ordinary differential equations of fractional order in a finite range of the real axis. The sixth chapter of the thesis provides a discussion and results.